Examples:
Here we demonstrate the approach by implementation with metarep, using examples from systematic reviews Cochrane library. These examples are detailed in the paper as well, along with a demonstration of a way to incorporate our suggestions in standard meta-analysis reporting system.
We demonstrate the with an example based on a fixed-effects meta-analysis. This example was included in Jaljuli et. al. 2020, found in review number CD002943 in the Cochrane library. This analysis explores the effect of mammogram invitation on attendance during the following 12 months.
library(metarep)
library(meta)
Loading 'meta' package (version 5.1-1).
Type 'help(meta)' for a brief overview.
Readers of 'Meta-Analysis with R (Use R!)' should install
older version of 'meta' package: https://tinyurl.com/dt4y5drs
Attaching package: 'meta'
The following object is masked from 'package:metarep':
forest
data(CD002943_CMP001)
m2943 <- metabin( event.e = N_EVENTS1, n.e = N_TOTAL1,
event.c = N_EVENTS2, n.c = N_TOTAL2,
studlab = STUDY, comb.fixed = T , comb.random = F,
method = 'Peto', sm = CD002943_CMP001$SM[1],
data = CD002943_CMP001)
Warning: Use argument 'fixed' instead of 'comb.fixed' (deprecated).
Warning: Use argument 'random' instead of 'comb.random' (deprecated).
m2943
Number of studies combined: k = 5
Number of observations: o = 4166
Number of events: e = 1384
OR 95%-CI z p-value
Common effect model 1.6564 [1.4317; 1.9164] 6.78 < 0.0001
Quantifying heterogeneity:
tau^2 = 0.1314 [0.0073; 1.6698]; tau = 0.3625 [0.0852; 1.2922]
I^2 = 70.3% [24.4%; 88.3%]; H = 1.84 [1.15; 2.93]
Test of heterogeneity:
Q d.f. p-value
13.47 4 0.0092
Details on meta-analytical method:
- Peto method
- Restricted maximum-likelihood estimator for tau^2
- Q-profile method for confidence interval of tau^2 and tau
summary(m2943)
OR 95%-CI %W(common)
Sutton-1994 1.2836 [0.9933; 1.6589] 32.3
Somkin-1997 1.8739 [1.5372; 2.2844] 54.2
Turnbull-1991 3.5709 [1.9225; 6.6326] 5.5
Mohler-1995 1.8764 [0.5272; 6.6788] 1.3
Bodiya-1999 1.0758 [0.6110; 1.8941] 6.6
Number of studies combined: k = 5
Number of observations: o = 4166
Number of events: e = 1384
OR 95%-CI z p-value
Common effect model 1.6564 [1.4317; 1.9164] 6.78 < 0.0001
Quantifying heterogeneity:
tau^2 = 0.1314 [0.0073; 1.6698]; tau = 0.3625 [0.0852; 1.2922]
I^2 = 70.3% [24.4%; 88.3%]; H = 1.84 [1.15; 2.93]
Test of heterogeneity:
Q d.f. p-value
13.47 4 0.0092
Details on meta-analytical method:
- Peto method
- Restricted maximum-likelihood estimator for tau^2
- Q-profile method for confidence interval of tau^2 and tauIn this meta-analysis, the effect of sending invitation letters was examined in five studies. The authors main result is that: “The odds ratio in relation to the outcome, attendance in response to the mammogram invitation during the 12 months after the invitation, was 1.66 (95% CI 1.43 to 1.92)”.
We suggest reporting the replicability-analysis results alongside: the r-value and lower confidence bounds on the number of studies. Results of complete replicability analysis can be added to the contents of a meta object or using the function metarep(...), as well as to its summary using summary( metarep(...) ).
To perform assumption-free replicability-analysis requiring replicability in at least u = 2 (default) studies, we calculate \(r(2)-value\) using truncated-Pearson’s’ test with truncation threshold t=0.05 (default):
(m2943.ra <- metarep(x = m2943 , u = 2 , common.effect = F ,t = 0.05 ,report.u.max = T))
Number of studies combined: k = 5
Number of observations: o = 4166
Number of events: e = 1384
OR 95%-CI z p-value
Common effect model 1.6564 [1.4317; 1.9164] 6.78 < 0.0001
Quantifying heterogeneity:
tau^2 = 0.1314 [0.0073; 1.6698]; tau = 0.3625 [0.0852; 1.2922]
I^2 = 70.3% [24.4%; 88.3%]; H = 1.84 [1.15; 2.93]
Test of heterogeneity:
Q d.f. p-value
13.47 4 0.0092
Details on meta-analytical method:
- Peto method
- Restricted maximum-likelihood estimator for tau^2
- Q-profile method for confidence interval of tau^2 and tauThe bottom two lines report the \(r(2)-value\), lower bound on the number of studies with increased effect (\(u^L_{max}\)) and decreased effect (\(u^R_{max}\)) , respectively. The evidence towards an increased effect was replicable, with \(r(2) − value = 0.0002\). Moreover, with \(95\%\) confidence, we can conclude that at least two studies had an increased effect. For higher replicability requirement, compute \(r(u')-value\) for \(u'>2\) using metarep(u = u' , ... ).
The two-sided \(r(u)-value\) of the model can be accessed via r.value:
m2943.ra$r.value
[1] 1.800734e-08The replicability-analysis reported was performed with an assumption free test, based on truncated-Pearson’s’ test with truncation level set at the nominal hypothesis testing level (i.e., t=0.05, default). For ordinary Pearson’s’ test, use t=1.
Although the fixed-effect model assumes that all studies are estimates of the same common effect \(\theta\), we recommend applying assumption-free replicability-analysis for protection against an (perhaps) unsupported assumption. Despite that, we extend our suggested method with the common-effect incorporation in section 7. This analysis can be performed via metarep( ... , common.effect = TRUE ).
metarep also allows adding replicability results to the conventional forest plots by meta. This can be done by simply applying forest() on a metarep object.
forest(m2943.ra, layout='revman5',digits.pval = 2 , test.overall = T )
The lower bounds \(u^L_{max} \, \text{and} \, u^R_{max}\) are calculated with \(1-\alpha = 95\%\) confidence level ( default), meaning that each of the null hypotheses \[H^{u^L_{max}/n}(L) \;\; \text{and}\;\; H^{u^L_{max}/n}(R)\] is tested at level \(\alpha /2 = 2.5\%\), resulting in bounds in overall type error rate \(5\%\). Type I error rate can be controlled for any desired \(\alpha\) using the argument confidence = 1 -\(\alpha\).
The calculation of \(u^L_{max} \, \text{and} \, u^R_{max}\) can also be calculated directly using the function find_umax() with the option to specify one-sided alternative, confidence level, truncation threshold and common-effect assumption. For example, let’s compute \(u^L_{max}\) with the same confidence level as produced by m2943.ra.bounds.
find_umax(x = m2943 , common.effect = F,alternative = 'less',t = 0.05,confidence = 0.975)
$worst.case
[1] "Sutton-1994" "Somkin-1997" "Turnbull-1991" "Mohler-1995"
[5] "Bodiya-1999"
$side
Direction of the stronger signal
"less"
$u_max
u^L
0
$r.value
r^R
1
$Replicability_Analysis
[1] "out of 5 studies, 0 with decreased effect."Note that this function produces 2 main types of results:
Worst-case scenario studies: A list of \(n-u_{max}^L+1\) studies names yielding the maximum \[\max_{\forall \{i_1,\dots , i_{n-u+1}\} \subset \{1,\dots , n\} } \{\,p^L_{i_1,\dots , i_{n-u+1}}\,\}\]
Replicability-analysis results, including:
\(u^L_{max}\) or \(u^R_{max}\). If
alternative='two-sided', then \(u_{max}=\max\{u^L_{max}\, , \, u^R_{max}\}\) is also reported.\(r(u_{max}^L)-value\) or \(r(u_{max}^R)-value\) if setting
alternative='less'oralternative='greater', respectively. Ifalternative='two-sided', then \[rvalue = r(u_{max}) = 2\cdot\min\{r^R(u_{max}^R) , r^L(u_{max}^L) \}\] is also reported.
For demonstration, see the following example.
find_umax(x = m2943 , common.effect = F,alternative = 'two-sided',t = 0.05,confidence = 0.95)
$worst.case
[1] "Sutton-1994" "Somkin-1997" "Mohler-1995" "Bodiya-1999"
$side
Direction of the stronger signal
"greater"
$u_max
u_max u^L u^R
2 0 2
$r.value
r.value r^L r^R
0 1 0
$Replicability_Analysis
[1] "out of 5 studies: 0 with decreased effect, and 2 with increased effect."Replicability analysis using a test-statistic based on
The common-effect assumption
find_umax(x = m2943 , common.effect = T,alternative = 'two-sided', confidence = 0.95)
Performing Replicability analysis with the common-effect assumption
$worst.case
[1] "Sutton-1994" "Mohler-1995" "Bodiya-1999"
$u_max
[1] 3
$side
[1] "greater"
$r.value
[1] 0.0468
(m2943.raFE <- metarep(x = m2943 , u = 2 , common.effect = T ,report.u.max = T))
Performing Replicability analysis with the common-effect assumption
Performing Replicability analysis with the common-effect assumption
Number of studies combined: k = 5
Number of observations: o = 4166
Number of events: e = 1384
OR 95%-CI z p-value
Common effect model 1.6564 [1.4317; 1.9164] 6.78 < 0.0001
Quantifying heterogeneity:
tau^2 = 0.1314 [0.0073; 1.6698]; tau = 0.3625 [0.0852; 1.2922]
I^2 = 70.3% [24.4%; 88.3%]; H = 1.84 [1.15; 2.93]
Test of heterogeneity:
Q d.f. p-value
13.47 4 0.0092
Details on meta-analytical method:
- Peto method
- Restricted maximum-likelihood estimator for tau^2
- Q-profile method for confidence interval of tau^2 and taufind_umax(x = m2943 , common.effect = T,alternative = 'two-sided', confidence = 0.95)
Performing Replicability analysis with the common-effect assumption
$worst.case
[1] "Sutton-1994" "Mohler-1995" "Bodiya-1999"
$u_max
[1] 3
$side
[1] "greater"
$r.value
[1] 0.0468
(m2943.raFE <- metarep(x = m2943 , u = 2 , common.effect = T ,report.u.max = T))
Performing Replicability analysis with the common-effect assumption
Performing Replicability analysis with the common-effect assumption
Number of studies combined: k = 5
Number of observations: o = 4166
Number of events: e = 1384
OR 95%-CI z p-value
Common effect model 1.6564 [1.4317; 1.9164] 6.78 < 0.0001
Quantifying heterogeneity:
tau^2 = 0.1314 [0.0073; 1.6698]; tau = 0.3625 [0.0852; 1.2922]
I^2 = 70.3% [24.4%; 88.3%]; H = 1.84 [1.15; 2.93]
Test of heterogeneity:
Q d.f. p-value
13.47 4 0.0092
Details on meta-analytical method:
- Peto method
- Restricted maximum-likelihood estimator for tau^2
- Q-profile method for confidence interval of tau^2 and tau
forest(m2943.raFE, layout='revman5',digits.pval = 2 , test.overall = T )